Department of Civil Engineering, University of Toronto35 St. George Street, Toronto, ON, M5S 1A4, Canadae-mail: fjv@civ.utoronto.ca

Key words: Fibre Reinforced Concrete, Steel Fibre, Tensile Stress, Crack Width, Bond, AnchorageAbstract: In order to represent the ductile tensile behaviour of steel fibre reinforced concrete(SFRC), the Diverse Embedment Model (DEM) was recently developed, accounting for both therandom distribution of fibres and the pull-out behaviour of fibres. Although the DEM shows goodagreement with test results measured from uniaxial tension tests, it entails a double numericalintegration which complicates its implementation into computational models and softwaredeveloped for the analysis of the structural behaviour of SFRC members.In this paper, the DEM is simplified by eliminating the double numerical integration; thus, theSimplified DEM (SDEM) is derived. In order to simplify the DEM, only fibre slip on the shorterembedded side is taken into the account of the fibre tensile stress at a crack, while coefficients forfrictional bond behaviour and mechanical anchorage effect are incorporated to preventoverestimation of the tensile stress attained by fibres due to the neglect of fibre slip on the longerembedded side. The tensile stress-crack width response of SFRC predicted by the SDEM showsgood agreement with that obtained from the DEM; hence, the model’s accuracy has largely beenretained despite the simplification. In comparisons with test results reported in the previousliterature, the SDEM is shown to simulate well not only the direct tensile behaviour but also theflexural behaviour of SFRC members. The SDEM can easily be implemented in currently availableanalysis models and programs so that it can be useful in the modelling of structural behaviour ofSFRC members or structures.

1 INTRODUCTIONIt is well known that steel fibre reinforcedconcrete (SFRC) exhibits a ductile post-cracking behaviour due to steel fibres bridgingcracks. Many researchers [1-4] investigatedthe beneficial aspect of SFRC in structuralmembers. However, SFRC is yet to be widelyapplied as a structural member in actualconstruction. One of the main reasons for thisis that most researches focused on qualitativeevaluations for the tensile behaviour of SFRC[5-9], rather than on the development of arational model which can be easily employedto predict the structural behaviour of SFRCmembers.Recently, several research groupsSeong‐Cheol Lee, Jae‐Yeol Cho and Frank J. Vecchio

2developed constitutive models for the tensilebehaviour of SFRC members. Marti et al. [10]developed a simple formula to predict thetensile stress-crack width relationship of SFRC,by assuming that the number of fibres bridginga crack decreases linearly with increasingcrack width. Later, an engagement factor toconsider the effect of fibre inclination angle onfibre pullout behaviour was introduced by Vooand Foster [11] who developed the VariableEngagement Model (VEM). Leutbecher andFehling [12] also presented a model thatconsiders the effect of fibres on crack widthsin SFRC members with conventionalreinforcing bars. Stroeven [13] developed aformulation that considered varying uniformbond stress along a fibre according to the fibretype. However, the appropriateness of thesemodels for SFRC members with end-hookedfibres is questionable because a uniform bondstress along a fibre is assumed.Recently, Lee et al. [14-15] proposed theDiverse Embedment Model (DEM) evaluatingthe tensile stresses due to the frictional bondbehaviour and the mechanical anchorage effectseparately so that the tensile behaviour ofSFRC with straight fibres or end-hooked fibrescould be accurately predicted. In the DEM,however, a double numerical integrationshould be undertaken in order to calculate theaverage tensile stress of steel fibres at a crack.This complicates the implementation of theDEM into various analysis models [16-19] andprograms [20-21] useful for the calculation ofthe structural behaviour of SFRC memberswith or without conventional reinforcing bars.In this paper, therefore, a simplified versionof the DEM (SDEM) will be derived byeliminating the double numerical integration inthe DEM by introducing some coefficientswithout significant loss of accuracy.2 DERIVATION OF THE SIMPLIFIEDDEM (SDEM)2.1 Fundamental assumptionIn the DEM formulation, with theassumption of a rigid body translation, thepullout behaviour of a single fibre embeddedon both sides can be analyzed, then theaverage tensile stress of fibres at a crack as thefollowing equation.,,2 2,0 02sinff cr avglfcr afd dll  

(1)where,fcris a fibre tensile stress at acrack, which is a function of the fibreorientation angle (⤠慮搠晩扲攠敭扥摭敮琠汥湧瑨 al). Although the DEM well predictsthe tensile behaviour of SFRC, the calculationof the average fibre tensile stress at a crack iscomplicating because of a double numericalintegration due to the compatibility conditionthat the crack width be equal to the sum of theslips on both sides of a fibre.In order to simplify the DEM, one moreassumption can be made with respect tocompatibility; the crack width can be assumedto be the same as the slip on the shorterembedded side while the slip on the longerembedded side is neglected. With thisassumption, the iteration procedure required toanalyze the pullout behaviour of a single fibreembedded on both sides can be omitted so thatthe double numerical integration in the DEMcan be averted. However, the effect of fibreslip of the longer embedded side on the fibretensile stress at a crack can be significant insome cases. Hence, in this paper, twocoefficients will be introduced within theformulation to compensate for the relaxedcompatibility condition. The details follow.2.2 Frictional bond behaviourIn the case of straight fibres, since it is

assumed that the slip of a fibre occurs only onthe shorter embedded side, a fibre tensile stressat a crack can be calculated by integrating thefrictional bond stress along the shorterembedment part of the fibre. In this paper, abilinear relationship between the bond stressand slip is employed for the frictional bondbehaviour of a fibre, as illustrated in Fig. 1,which considers the effect of fibre inclinationangle on the frictional bond behaviour. Thefrictional bond strength is constant while theSeong‐Cheol Lee, Jae‐Yeol Cho and Frank J. Vecchio

3slip at the peak increases with an increase ofthe fibre inclination angle, as assumed in theDEM based on test results reported by Banthiaand Trottier [23]. Note that the slip reported inthe figure is the same as the crack widthbecause the slip of a fibre on the longerembedded side is neglected.

Figure 1: Frictional bond behaviour of a single fibre[22].Since a bilinear relationship is employedfor the frictional bond behaviour, two phasesshould be considered in the calculation of thefibre tensile stress at a crack. The first occurswhen the crack width is so small that all fibresare still on the linearly ascending part of theconstitutive law for the frictional bondbehaviour; the second prevails when the crackwidth is large such that some fibres exhibitplastic frictional bond behaviour while otherfibres remain in the pre-peak regime.Without suitable compensation made, thefibre tensile stress can be significantlyoverestimated when the fibre slip on the longerembedded side is neglected, particularly for afibre which does not reach the frictional bondstrength. This effect of a fibre slip on thelonger embedded side quickly diminishes aftera fibre reaches the frictional bond strength,because the slip on the longer embedded sidedecreases as the fibre tensile stress decreaseswith an increase in the crack width. Therefore,in order to consider the effect of slip of thefibre on the longer embedded side on thefrictional bond stress of a fibre, a factor,fⰠ睩汬⁢攠慰灬楥搠瑯⁦楢牥w ⁮潴⁨慶楮朠牥慣桥搠瑨攠晲楣瑩潮慬⁢潮搠獴牥湧瑨⁷桥渠瑨攠慶敲慧攠晲楣瑩潮慬⁢潮搠獴牥獳⁯爠瑨攠慶敲慧攠晩扲攠瑥湳楬攠獴牥獳⁩猠捡汣畬慴敤⸠䙯爠瑨攠晩牳琠灨慳攠潦⁲敳灯湳攠楮⁷桩捨⁴桥F捲慣欠睩摴栠楳⁳c慬汥爠瑨慮⁴桥⁳汩 fscorresponding to the initiation of plasticfrictional bond behaviour of a fibreperpendicular to the crack surface, the averagefrictional bond stress considering the randomdistribution of the fibre inclination angle canbe calculated as follows:,,max3fcrf avg ffws  forcr fw s

Figure 2: Comparison of SDEM with DEM for straightfibres.2.3 Mechanical anchorage effectInthe case of end-hooked fibres, the effectof mechanical anchorage on the pulloutbehaviour should be considered in addition tothe frictional bond behaviour. From the testresults presented by Banthia and Trottier [23],the effect of fibre inclination angle on themechanical anchorage effect can be assumedto be the same as for straight fibres; themaximum force due to the mechanicalanchorage is constant while the slip at the peakincreases with an increase in the fibreinclination angle. Based on the work ofSujivorakul et al. [25], the relationshipbetween fibre slip and tensile force due to themechanical anchorage is idealized withparabolic and linear relationships for the pre-and post- peak behaviours, respectively, withconsideration of the fibre inclination angleeffect as illustrated in Fig. 3 [14].

Figure 3: Mechanical anchorage behaviour in an end‐hooked fibre [14].Similar to the frictional bond behaviour,three phases can be considered in thecalculation of the fibre tensile stress due tomechanical anchorage; pre-peak, post-peak,and full deterioration of an end-hook. Beforethe beginning of the full deterioration of anend-hook, through the same procedurepresented for the frictional bond behaviour, theaverage tensile force due to mechanicalanchorage can be calculated withconsideration given to the random distributionof the fibre inclination angle as follows:2,,max2 13 5cr creh avg eh eheh ehw wP Ps s        

forcr ehw s(7) ,2,max71 1152

eh eheh avgcrcr ehehf isPww sPl l     

for2fieh crl ls w 

(8)Seong‐Cheol Lee, Jae‐Yeol Cho and Frank J. Vecchio

5When the crack width is sufficiently largeto cause some of end-hooks to fully deteriorate,the equation to evaluate the average tensileforce due to mechanical anchorage becomestoo difficult to derive exactly throughintegration. Therefore, a simple parabolicrelationship between the crack width and theaverage tensile force caused by mechanicalanchorage can be employed as follows:2,,,22i creh avg eh avg ii fl wP Pl l    

for2 2f iicrl llw  (9)where,,eh avg iPis the average tensile forcedue to the mechanical anchorage at 2cr f iw l l calculated from Eq. (8).When the crack width is larger than2il, itcan be assumed that all mechanical anchorageshave fully pulled out.In the calculation of the average fibretensile stress at a crack due to the mechanicalanchorage effect, the fibres in which themechanical anchorage has pulled out shouldnot be considered. Therefore, assuming auniform distribution over the shorterembedment length of fibres at initial cracking,the fibre tensile stress at a crack due to themechanical anchorage effect can be calculatedas follows:,,,242eh avgi crf cr ehf fPl wd l(10)By introducing the maximum bond strengthdue to the mechanical anchorage of an endhooked fibre,max,max2eh eh f fP d l, thetensile stress of an SFRC element due to themechanical anchorage effect can be calculatedas follows:,max2 2i creh f f eh ehfl wf V Kd (11)whereehKis referred to Eqs. (7)~(9).Finally, the tensile stress attained in SFRCelements with end-hooked fibres can becalculated from the superposition of the tensilestresses due to the frictional bond behaviourand the mechanical anchorage effect. Fig. 4compares the tensile stress attained by end-hooked fibres as calculated by DEM andSDEM. It can be seen that the results of thesimplified model show good correspondencewith the DEM.0.01.02.03.04.00.0 1.0 2.0 3.0 4.0Stress due to fibres (MPa)Crack width (mm)DEMSDEMVf=1.5%lf=30mm; df=0.5mmeh,max=3.69 MPaf,max=3.46 MPaseh=0.01 mm0.05 mm0.10 mm0.50 mm

Figure 4: Comparison of SDEM with DEM for end‐hooked fibres.2.4 Tensile stress of SFRCThe formulations above have dealt with thetensile stress attained by steel fibres. Toevaluate realistically the tensile stress responseof SFRC members, the tensile stress due to thetension softening effect of concrete matrixshould be added to that attained by steel fibres.This study adopted the following exponentialform [11] for the tension softening effect.crcwct crf f e

(12)where the coefficientcis 15 and 30 forconcrete and mortar, respectively.Therefore, the tensile stress of a SFRCmember can be calculated as follows:SFRC f ctff f

63 VERIFICATION OF SDEM3.1 Uniaxial tensile behaviour of SFRCFor the verification of the proposed model,the predictions of SDEM were compared withexperimental data obtained from otherresearchers’ investigations [5,26]. The testresults were also compared with thepredictions of other researchers’ proposedmodels [10-13]. When the SDEM wasemployed to evaluate the tensile stress attainedby steel fibres, the slips corresponding to thebond strength due to the frictional bondbehaviour,fs, and the maximum force due tothe mechanical anchorage,ehs, were assumedto 0.01 and 0.1 mm, respectively, as suggestedby Naaman and Najm [27]. The frictional bondstrength,,maxf, and the mechanical anchoragestrength,,maxeh, were assumed to be'0.396cfand'0.429cf, respectively, basedon the previous studies [11,15].As compared in Fig. 5~6, the SDEM showsthe best agreement with the test results notonly for the specimens with straight fibres butalso for the specimens with end-hooked fibres.This is primarily due to differences in thefundamental assumptions; the SDEMconsiders both the frictional bond behaviourand the mechanical anchorage effectseparately, whereas the other models assumesconstant bond stress along fibres even for end-hooked fibres. Therefore, it can be concludedthat the structural behaviour of SFRCmembers subjected to direct tension can beaccurately represented by the SDEM.

Figure 6: Comparison for the members with end‐hooked fibres tested by Susetyo [26].

3.2 Flexural behaviour of SFRCTo investigate the modelling capabilities ofthe SDEM for flexural members, the four-point bending tests were considered. In theanalysis of the flexural behaviour of SFRCspecimens, the sectional analysis procedurepresented by Oh et al. [28] was employed.In the flexural analysis, it was assumedthat a SFRC beam specimen subjected to theSeong‐Cheol Lee, Jae‐Yeol Cho and Frank J. Vecchio

8four-point loading reaches failure through theformation of a single dominant flexural crack,as presented in Fig. 7. From the geometriccondition illustrated in this figure, therelationship between the compressive strain ofthe top fibre in the pure bending region andthe centre deflection can be derived. Then, asillustrated in Fig. 8, the stress distributionalong the section with a flexural crack can beseparately evaluated for the un-cracked depthwith the strain distribution and the crackeddepth with the crack width distribution.Consequently, the sectional analysis for asection with a flexural crack can be conducted.

Figure 7: Failure model of a SFRC beam with asingle dominant crack [28].As a verification of the SDEM, the flexuralspecimens tested by Susetyo [26] wereanalyzed. As shown in Fig. 9, the analysisresults obtained from the SDEM show goodagreement with the test results for the flexuralbehaviour of the SFRC members.

9the simplification, it was assumed that thefibre slip on the shorter embedded side is thesame as the crack width. As a result, the fibretensile stress at a crack can be calculateddirectly for a given crack width byconsidering the same constitutive models forfrictional bond behaviour and the mechanicalanchorage effect as employed in the DEM. Toprevent an overestimation of the fibre tensilestress caused by neglecting the effect of afibre slip on the longer embedded side, thecoefficients,f⁡湤ehwere introduced forthe frictional bond behaviour and themechanical anchorage effect, respectively.Consequently, the tensile stress attained byfibres in SFRC members can be more simplyevaluated.The accuracy of the SDEM was verifiedthrough the analysis of various test specimens.The tensile stress-crack width responses ofSFRC calculated by the SDEM showed goodagreement with those obtained from the DEM.In comparisons with test results, the SDEMpredicted well the direct tensile behaviour ofSFRC members with straight fibres or end-hooked fibres. From sectional analyses withthe failure mode exhibiting a single dominantflexural crack, the SDEM showed also goodagreement with the test results for the flexuralbehaviour of SFRC beams. Consequently, itcan be concluded that the tensile or flexuralbehaviour of SFRC members can be modelledsimply and accurately with the SDEM.The proposed SDEM can be easilyimplemented into currently available analysismodels [19-22] or programs [23-24] so that itcan be useful in the assessment of thestructural behaviour of SFRC members orstructures with or without conventionalreinforcing bars.ACKNOWLEDGEMENTThis research was partially supported by“Basic Science Research Program through theNational Research Foundation of Korea(NRF) (20120003756)” funded by theMinistry of Education, Science andTechnology.REFERENCES[1] Parra-Montesinos, G.J., 2005. High-Performance Fiber-Reinforced CementComposites: An Alternative for SeismicDesign of Structures.ACI Struct. J.